## Abstract

The expression of genes in the cell is controlled by a complex interaction network involving proteins, RNA and DNA. The molecular events associated with the nodes of such a network take place on a variety of time scales, and thus cannot be regarded as instantaneous. In many cases, the cell is robust with respect to the delay in gene expression control, behaving as if it were instantaneous. However, there are specific cases in which delay gives rise to temporal oscillations. This is the case, for example, of the expression of tumour-suppressor protein p53, of protein Hes1, involved in the differentiation of stem cells, of NFkB and Wnt, in which case delay arises implicitly from the structure of the associated network. By means of delay rate equations, we study the kinetics of small regulatory networks, emphasizing the role of delay in an evolutionary context. These models suggest that oscillations are a typical outcome of the dynamics of regulatory networks, and evolution has to work to avoid them when not required (and not vice versa).

## 1. Introduction

The life of any cell is regulated by the timely production and circulation of the macromolecules (proteins and RNA) encoded in its DNA. The appearance in the nucleus of a given protein activates a gene, which expresses another protein, which causes the degradation of still another protein, and so on. The result is a complicated system of interactions among biological macromolecules. The coupled dynamics of such macromolecules determines the metabolism of the cell. In other words, it controls and elaborates the flow of information whose input is the set of external stimuli that act on the cell and whose output is its macroscopic behaviour.

In order to give a physical description of such complex processes, it is necessary to resort to approximations. Because macromolecular diffusion within each compartment is one of the fastest processes taking place in cellular metabolism, one can approximate the associated concentration as uniform, and specify the state of the cell at a given time simply in terms of such concentrations, neglecting the motion of single molecules. If the number of molecules is large enough, then one can express the concentration of a given macromolecule in a given cellular compartment as a real number *x*_{i}.

The mutual effect of these proteins and RNA can thus be summarized by a network, where each node represents the concentration *x*_{i} and each link represents the mutual effect between macromolecules, such as activation, inhibition, degradation, translocation, etc. (figure 1).

The simplest way to describe the dynamics of the network is by means of deterministic rate equations for the concentrations *x*_{i}, such as
1.1where the function *g*_{ij} accounts for the effect of molecule *i* on molecule *j*, and the constant *g*_{i} is the spontaneous degradation rate of the *i*th molecule. Common forms of *g*_{ij}(*x*_{i}) are the linear function and a step function , used when molecule *j* is activated (*a*>0) or degraded (*a*<0) by the binding of *h* molecules of kind *i*.

An implicit assumption behind equations (1.1) is that the effect of molecule *j* on molecule *i* can be regarded as instantaneous. This is seldom a realistic assumption. The cellular processes summarized by the functions *g*_{ij} involve a set of time scales that range from milliseconds to days. For example, protein translocation through nuclear pores takes place in 10^{−4} s [1], translation lasts for about 30 s, transcription and mRNA degradation occurs in about 3 min [2], whereas protein degradation takes from minutes to hours.

As a consequence of this wide range of time scales, the effect of one molecule on another one must often be regarded as delayed, and the associated rate equation must be expressed as delay differential equations such as 1.2

A common effect of delay in the rate equations is the uprise of temporal oscillations in the associated quantities [3]. In fact, oscillations in protein concentration have been observed in a number of important systems (figure 2), such as the tumour-suppressor protein p53 [4], proteins involved in embryonal somite segmentation such as Hes1, Axin, Notch, Wnt [6,9] and in immune response as NFkB [7]. In all these cases, the very physical reason for time oscillations was found to be a delay in some elements of the control network of these proteins [10–12] (for a review, see refs. [13,14]).

In some cases, the cell benefits from the oscillating behaviour of its proteins. For example, time oscillations of Hes1 are turned into spatially repetitive units such as vertebrae, ribs and skeletal muscles [15]. Oscillations in p53 are thought to coordinate the cellular response to DNA damage, causing growth arrest and eventually apoptosis [16]. In other cases, the cellular metabolism requires a steady response, and oscillations are a nuisance to the correct functioning of the cell. An important question one can ask concerns what is the evolutionary role of temporal oscillations in the concentration of macromolecules. One can envisage two limiting scenarios. In the former, oscillations are a rare phenomenon that takes place in exceptional conditions, and evolution has put a strong effort to obtain it. In the opposite scenario, oscillations are the normal effect of the complexity of the genetic network, and evolution has developed countermeasures to suppress them when not required.

Physical models of genetic networks can help to elucidate the earlier mentioned issue. In the following sections, we discuss some simple cases of delayed genetic response, trying to distinguish between the necessary outcomes of a typical delay system and the effects that required a fine-tuning of the underlying parameters.

## 2. The simplest genetic-control element with delay

A simple model one can study is that of a molecule that inhibits itself, such as that sketched in figure 1*a*. We will discuss it in detail because it allows one to capture the main point associated with a delay response, summarizing in the next sections numerical results obtained with more complex systems.

Although, in the cell, the smallest control element should involve at least two molecules, namely a protein and its mRNA, one can give an effective description of the mRNA and accounting explicitly for the protein only. In this case, the effective delay is the sum of that associated with transcription and translation. Of course, the element displayed in figure 1*a* can also approximate more complex systems, involving more than two molecules, provided that the delay and the functional form of the rate equations account correctly for the whole system.

In the simplest case, one can study a delay linearized system in the form
2.1where *S* is a constant rate of protein production, *a* is its spontaneous degradation rate and *b* is the induced degradation rate displaying a delay *τ*. The general solution of equation (2.1) is , which gives
2.2The system displays sustained oscillations when Re *λ*=0, which gives
2.3and
2.4A more general treatment [17] gives the conditions under which the system converges to a stationary solution, that is |*a*|>|*b*| for any *τ* or if |*a*|<|*b*| for . Roughly speaking, the system can oscillate if the dominant part of the kernel is the delayed one and if the delay is large enough. The associated phase diagram is displayed in figure 3. If the effect of the delayed degradation is not negligible (*b*≫*a*), then the value of the delay above which the system experiences oscillations is a fraction of *a*^{−1}. For example, if one assumes that instantaneous degradation *a*∼1 h^{−1} reflects spontaneous degradation, whereas delay degradation *b*=10*a* is that under genetic control and consequently much larger than *a* (this is somewhat tautological: if degradation is controlled, then it should overwhelm spontaneous degradation), then the threshold value of *τ* is 9 min. This threshold is comparable with the typical delay associated with the molecular processes summarized by equation (2.1).

This simple model suggests that the range of parameters that define genetic control elements display a consistent probability to be found in the oscillatory region. In this case, oscillations can be regarded as the typical outcome of a genetic control dynamics, whereas evolution must put an effort to avoid them when not required.

## 3. Dynamics of some important biological control elements with delay

The protein Hes1 has been shown to display regular temporal oscillations [6], which are involved in the formation of repetitive structures, such as vertebrae, ribs and skeletal muscles in embryos. The oscillation period has been shown to be constant in each species [18] (30 min in zebrafish, 90 min in chicks, 2 h in mice, 8 h in humans). This system is particularly simple, because oscillations are apparently controlled only by a two-species system in which Hes1 represses the transcription of its own mRNA [6] (figure 1*b*), and no post-translational modification of the protein is required [18].

A simple model to account for these oscillations is described in [11], making use of the delay rate equations
3.1where *x* is the concentration of the protein Hes1 and *y* of its mRNA. Because Hes1 acts on its mRNA as a dimer, *h* is set to 2, whereas *k*_{m}=0.04 min^{−1}, *k*_{p}=0.04 min^{−1} [6], and the values of the other parameters (which are not known experimentally in detail) are set to the typical values *α*=*β*=1 min^{−1} and *k*=0.1.

A numerical analysis of the solution [11] of these equations shows that for a wide range of delays, i.e. , the system displays oscillations with a fixed period of 170 min. For *τ*<10 min, no oscillations are observed, whereas for *τ*>60 min, the oscillation period increases. Variations of *α*, *β* and *k* of five orders of magnitude around their basal values cause no qualitative difference in the oscillations. Only the increase of *k*_{m} and *k*_{p} disrupts the oscillatory mechanism. Summing up, oscillations appear rather robust with respect to most of the controlling parameters, including the time delay.

The first and most studied system displaying oscillations is that built out of proteins p53 and mdm2 [4]. This system is activated by DNA damage, and the peaks in p53 concentration activate a cascade of events, which lead to growth arrest and apoptosis. Protein p53 is produced at constant rate and activates the expression of mdm2, which, in turn, binds to p53, causing its degradation. Consequently, under normal cellular conditions, the concentration of p53 is kept low. If cellular DNA is damaged, then various types of kinases are produced to phosphorylate the region of p53 that binds mdm2, resulting in sustained peaks in p53 concentration. About 50% off all tumours are associated with a failure of this mechanism [19].

Single-molecule experiments have shown that p53 oscillations following gamma-ray damage to DNA last for at least 72 h and their period does not depend on the intensity of the gamma rays [5,20].

The first description of the p53 system was done through the delay rate equations
3.2where *x* is the concentration of p53, *y* that of mdm2 and *z* that of their complex. According to the first line, p53 is produced at constant rate *S*, displays a spontaneous degradation with rate *b* and is degraded with rate *a* when bound to mdm2. On the other hand, mdm2 is produced at rate *c* when p53 is bound to its activation site on the DNA, the binding constant being *k*, whereas it is degraded spontaneously with rate *d*.

A numerical analysis of equation (3.2) shows that oscillations are robust for variations of all the parameters of the system except when *d* or *k*_{g} are consistently decreased. A careful analysis of how the dynamics of the protein concentration depends on the parameters of the rate equations has been carried out by Neamţu and co-workers [21]. The main conclusion of this work is that under the condition that the dissociation constant *k* between p53 and mdm2 is small, there exists a critical delay above which the system shows oscillations. Again, oscillations seem to be remarkably robust with respect to the parameters defining the system.

## 4. Systems with implicit delay

In the case of control elements composed of more than two species, an effective delay can appear due to the slow dynamics of one of the species. The simplest example of this behaviour is the ‘repressilator’ [22], which is a synthetic system built out of three proteins repressing each other as displayed in figure 1*d*. This can be described by the set of non-delay rate equations
4.1where *y*_{i} and *x*_{i} (*i*=1,2,3) are the concentrations of the three proteins and of the associated mRNA, respectively. The interaction between any two proteins of the system (say, 1 and 3) is mediated by the other one (say, 2), which reacts to the concentration variations in a finite characteristic time, thus introducing an effective delay. A numerical analysis of equation (4.1) indicates that, for values of *α* larger than approximately 10^{1}, the steady-state solution is unstable, and oscillating behaviour appears (cf. fig. 1*b* in [22]). Of course, one can also introduce an explicit delay in equation (4.1), making the appearance of oscillations even more likely [23].

A very important genetic control element is that controlling the expression of protein NFkB, which is involved in a variety of cellular processes, including immune response, inflammation and development. The nuclear import and export of NFkB is controlled by another protein, IkB, whose expression is activated by NFkB. Under specific conditions, nuclear/cytoplasmic NFkB has been observed to undergo time oscillations [7,8], whereas its total amount is almost constant on the experimental time scale.

The oscillations of nuclear/cytoplasmic NFkB have been well described in terms of shape, phase, time period and frequency response by the rate equations [12] (figure 1*e*)
4.2where *x* is nuclear NFkB, *y* is cytoplasmic IkB and *z* is IkB mRNA. These equations produce sustained oscillations in all variables over a wide range of parameter values, provided that *ϵ* is small. In fact, in this case, the degradation rate of IkB is independent of *z*(*t*) (‘saturated degradation’), thus introducing an effective time delay into the feedback loop: it allows IkB to accumulate and stay around much longer than in the case of non-saturated degradation. In addition, in this case, the effective delay results in time oscillations of protein concentrations for a large set of parameters [12].

A similar behaviour is followed by Wnt, which builds out a negative feedback loop together with β-catenin and Axin2. In addition, in this case, the system displays oscillations as a consequence of an effective time delay in the negative response mechanism, delay arising from the complexity of the control mechanism. Even more interesting, oscillations are remarkably robust with respect to the parameters that define the model [24].

## 5. Amount of information in genetic control elements

The above-mentioned discussion suggests that it is quite natural for genetic control elements to generate oscillating responses. A related issue is whether the cell makes some gain by this type of response, with respect, for example, to a steady-state response.

The goal of genetic control elements is to process input signals, coming from cellular sensors or from other elements, to generate suitable output signals. In this respect, a cell acts similarly to a computer. Evolution has generated such elements to act efficiently enough to guarantee the correct functioning of the cell. An estimator of the quality of a control element is then the amount of information that it can elaborate.

Consider a simple control element, described by some set of rate equations, which gives a steady sigmoidal response *y* with respect to the concentration of some protein *x*. In principle, for each value of input *x*, there is a unique output *y*. The information capacity of this system thus seems infinite, because any real value between zero and the maximum of the sigmoidal function can be addressed by an appropriate choice of *x*. Of course, this is not true in real life, and is a consequence of the approximations carried out to describe the system by means of rate equations.

First, the number of proteins in a cell is described by integers, which are likely to be limited due to the finite amount of molecules and energy available. Consequently, the above-mentioned estimate, based on real numbers, is certainly too optimistic. Moreover, rate equations neglect the physical fact that all the processes involved in genetic control are affected by noise. As shown by Bialek and co-workers [25], a noisy system displaying sigmoidal response and controlled by realistic parameters can give only a two-state response. In terms of Shannon's information entropy, its capacity is bit.

Systems with delay can be treated mathematically including noise. In this case, rate equations are not useful, and one has to resort to more sophisticated tools. For example, Bratsun *et al*. [26] describe the variation of the number of protein molecules as a one-step stochastic process with delay, under the approximation that the delay is large enough to produce decorrelated quantities. In this case, they show that the system can produce time oscillations even in that region of parameter space where the corresponding deterministic system would display a steady state.

The information capacity of oscillating systems affected by noise can be estimated by Shannon's theorem [27]
5.1where *T* is the duration of the oscillations, *W* is the maximum frequency in its spectrum, *P* is the power of the spectrum and *N* is the average noise power. In the examples discussed above, the oscillations last for several hours, so we set *T*≈10 h. We do not expect in the spectrum any frequency higher than the inverse of the fastest process taking place in the generating genetic control elements, so *W*≈10^{−1} min^{−1}. The signal-to-noise ratio *P*/*N* is difficult to quantify. However, one can assume that the order of magnitude of the noise is similar to that of the signal, and consequently *P*≈*N*, giving . This means that, even when affected by noise, the information capacity of a genetic control element displaying oscillations is much larger than that of a stationary one. Of course, the information capacity is the maximum information the element can store, and it is not trivial to study how the cell can exploit it. Anyway, the number of bits that can be stored in an oscillating system seems much larger than that of a stationary system.

## 6. Conclusions

In the past decade, many genetic control systems have been shown to display oscillating response to stimuli. The role of these oscillations has been extensively questioned. In particular, it is not clear how difficult it is for the cell to produce temporal oscillations. In other words, it is not clear if a realistic control element will naturally produce oscillations, or if evolution has to design a very specific system to obtain an oscillating behaviour.

The physical analysis of genetic control elements carried out by means of rate equations indicates that the basic ingredient that causes oscillations in the concentration of a macromolecule in the cell is a delay in the system that controls the expression of that macromolecule. Owing to the wide range of time scales involved in the biological phenomena that control protein expression, time delay inevitably affects all control elements. Time delay in small control systems (those built out of few elements, such as a protein and its mRNA only) is already sufficient to produce sustained oscillations. The larger the genetic control element, the wider is the range of time scales involved and consequently the delay in the response. The robustness of the delay with respect to the controlling parameters that we have found in the calculations contributes to suggest that oscillations are not a rare effect that the system displays under extreme conditions. On the contrary, from an evolutionary point of view, oscillations are expected to be a common response to stimuli. Conversely, a steady-state response in a delay system is something exceptional, which is expected to be finely tuned by evolution to avoid oscillations.

We can think of genetic control elements as biological computers whose goal is to elaborate input stimuli to obtain an appropriate output. For example, transcriptional inhibition can be regarded as a logical ‘not’ node in the cellular computing network, whereas promoters can build out logical ‘and’ nodes. Cellular life can thus be summarized in a continuous elaboration and diffusion of signals (besides the simpler production of energy, required for the former goal). From this perspective, the higher the rate of transmission of information, the more efficiently can the cell elaborate reactions to external stimuli, similar to serial buses in electronic computers. Consequently, the likelihood of oscillations in the cellular response can be beneficial to the cell. In fact, the information capacity of a system that produces an oscillating response is much larger than that which can produce a stationary response.

However, there is a threat to the above-described mechanism. Complicated cellular networks can, in principle, undergo bifurcations to a chaotic regime. By definition, a chaotic dynamics would spoil the ability of the cell to elaborate and transmit information, and most probably would lead to disastrous consequences. In the numerical study of small genetic systems, extracted from more complex networks, we have never observed chaotic dynamics. However, low-dimensional systems are less likely to enter a chaotic regime than high-dimensional ones, as suggested by the Poincaré–Bendixson theorem. Consequently, one cannot exclude the possibility that specific choices of the parameters of the network, or accounting for more nodes in the network, can result in the finding of chaotic behaviour.

Thus, a future challenge for the investigation of the dynamics of cellular control elements is to understand if a chaotic regime is possible, or if evolution has set countermeasures to try to avoid it. Losing control of its metabolism is worse than death for the cell. The cell has developed several controlled ways of undergoing apoptosis, and the failure of them is usually associated, in complex organisms, with the uprise of tumours. Among the several perspectives connected with the study of the dynamics of cellular networks is then the understanding of the relation between chaotic regime and tumours.

## Footnotes

One contribution of 15 to a Theme Issue ‘Dynamics, control and information in delay-coupled systems’.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.